Chapter 10.5, Problem 40AYU

In Problems 31-42, rotate the axes so that the new equation contains no $xy$-term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40.

$34x^2-24xy+41y^2-25=0$

To determine

a.

The new equation that contains no $xy$ term.

Answer to Problem 40AYU

${x^{\prime }}^2+\frac{{y^{\prime }}^2}{\frac{1}{2}}=1$

Explanation of Solution

Given:

$34x^2-24xy+41y^2-25=0$

Calculation:

Consider the given equation $34x^2-24xy+41y^2-25=0$.

Using problem 40, we find that $\theta \approx 36.9°$, $x=\frac{1}{5}\left(4x^{\prime }-3y^{\prime }\right)$, $y=\frac{1}{5}\left(3x^{\prime }+4y^{\prime }\right)$.

Substituting $x$, $y$ in the above equation we get,

$34{\left(\frac{1}{5}\left(4x^{\prime }-3y^{\prime }\right)\right)}^2-24(\frac{1}{5}\left(4x^{\prime }-3y^{\prime }\right)\frac{1}{5}\left(3x^{\prime }+4y^{\prime }\right))+41{\left(\frac{1}{5}\left(3x^{\prime }+4y^{\prime }\right)\right)}^2-25=0$

$34\left(\frac{1}{25}\left(16{x^{\prime }}^2+9{y^{\prime }}^2-24x^{\prime }{y}^{\prime }\right)\right)-\frac{24}{25}\left(12{x^{\prime }}^2-12{y^{\prime }}^2+7x^{\prime }{y}^{\prime }\right)+\frac{41}{25}\left(9{x^{\prime }}^2+16{y^{\prime }}^2+24x^{\prime }{y}^{\prime }\right)-25=0$

$544{x^{\prime }}^2-816x^{\prime }{y}^{\prime }+306{y^{\prime }}^2-288{x^{\prime }}^2-168x^{\prime }{y}^{\prime }+288{y^{\prime }}^2+369{x^{\prime }}^2+984x^{\prime }{y}^{\prime }+656{y^{\prime }}^2-625=0$

$25{x^{\prime }}^2+50{y^{\prime }}^2=25$

${x^{\prime }}^2+2{y^{\prime }}^2=1$

${x^{\prime }}^2+\frac{{y^{\prime }}^2}{\frac{1}{2}}=1$

Ellipse; Center is at $\left(0, 0\right)$, Major axis is the $x^{\prime }\text{-axis}$, vertices at $\left(\pm 1,0\right)$.

To determine

b.

To graph: ${x^{\prime }}^2+\frac{{y^{\prime }}^2}{\frac{1}{2}}=1$ in $x^{\prime }{y}^{\prime }\text{-plane}$.

Answer to Problem 40AYU

Explanation of Solution

Given:

$34x^2-24xy+41y^2-25=0$

Graph:

The graph of ${x^{\prime }}^2+\frac{{y^{\prime }}^2}{\frac{1}{2}}=1$ in $x^{\prime }{y}^{\prime }\text{-plane}$.

The graph of $34x^2-24xy+41y^2-25=0$ which is also the graph of ${x^{\prime }}^2+\frac{{y^{\prime }}^2}{\frac{1}{2}}=1$ in $x^{\prime }{y}^{\prime }\text{-plane}$.

It is an ellipse; Center is at $\left(0, 0\right)$, Major axis is the $x^{\prime }\text{-axis}$, vertices at $\left(\pm 1,0\right)$.